Edexcel P3 (Pure Mathematics 3) 2018 Specimen

1. Express

$$\frac { 6 x + 4 } { 9 x ^ { 2 } - 4 } - \frac { 2 } { 3 x + 1 }$$ as a single fraction in its simplest form.

2. $$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 4 x - 12$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as $$x = \sqrt { \left( \frac { 4 ( 3 - x ) } { ( 3 + x ) } \right) } \quad x \neq - 3$$ The equation \(x ^ { 3 } + 3 x ^ { 2 } + 4 x - 12 = 0\) has a single root which is between 1 and 2
2. Use the iteration formula $$x _ { n + 1 } = \sqrt { \left( \frac { 4 \left( 3 - x _ { n } \right) } { \left( 3 + x _ { n } \right) } \right) } \quad n \geqslant 0$$ with \(x _ { 0 } = 1\) to find, to 2 decimal places, the value of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) The root of \(\mathrm { f } ( x ) = 0\) is \(\alpha\).
3. By choosing a suitable interval, prove that \(\alpha = 1.272\) to 3 decimal places.
   

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3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the graph \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = 2 | 3 - x | + 5 \quad x \geqslant 0$$

1. Solve the equation $$f ( x ) = \frac { 1 } { 2 } x + 30$$ Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has two distinct roots,
2. state the set of possible values for \(k\).
   

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4. (i) Find $$\int _ { 5 } ^ { 13 } \frac { 1 } { ( 2 x - 1 ) } \mathrm { d } x$$ writing your answer in its simplest form.
(ii) Use integration to find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin 2 x + \sec \frac { 1 } { 3 } x \tan \frac { 1 } { 3 } x \mathrm {~d} x$$

5. Given that $$y = \frac { 5 x ^ { 2 } - 10 x + 9 } { ( x - 1 ) ^ { 2 } } \quad x \neq 1$$ show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k } { ( x - 1 ) ^ { 3 } }\), where \(k\) is a constant to be found.
(6)

1. The functions f and g are defined by

$$\mathrm { f } : x \mapsto \mathrm { e } ^ { x } + 2 \quad x \in \mathbb { R }$$ $$\mathrm { g } : x \mapsto \ln x \quad x > 0$$

1. State the range of f .
2. Find \(\mathrm { fg } ( x )\), giving \(y\) our answer in its simplest form.
3. Find the exact value of \(x\) for which \(\mathrm { f } ( 2 x + 3 ) = 6\)
4. Find \(\mathrm { f } ^ { - 1 }\) stating its domain.
5. On the same axes sketch the curves with equation \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), giving the coordinates of all the points where the curves cross the axes.

1. The point \(P\) lies on the curve with equation

$$x = ( 4 y - \sin 2 y ) ^ { 2 }$$ Given that \(P\) has \(( x , y )\) coordinates \(\left( p , \frac { \pi } { 2 } \right)\), where \(p\) is a constant,

1. find the exact value of \(p\) The tangent to the curve at \(P\) cuts the \(y\)-axis at the point \(A\).
2. Use calculus to find the coordinates of \(A\).

8. In a controlled experiment, the number of microbes, \(N\), present in a culture \(T\) days after the start of the experiment were counted.
\(N\) and \(T\) are expected to satisfy a relationship of the form $$N = a T ^ { b } \quad \text { where } a \text { and } b \text { are constants }$$

1. Show that this relationship can be expressed in the form $$\log _ { 10 } N = m \log _ { 10 } T + c$$ giving \(m\) and \(c\) in terms of the constants \(a\) and/or \(b\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d8e25332-3a45-43ca-a5b8-0a16f47f13b9-24_1223_1043_895_461} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows the line of best fit for values of \(\log _ { 10 } N\) plotted against values of \(\log _ { 10 } T\)
2. Use the information provided to estimate the number of microbes present in the culture 3 days after the start of the experiment.
3. With reference to the model, interpret the value of the constant \(a\).

9. (a) Prove that $$\sec 2 A + \tan 2 A \equiv \frac { \cos A + \sin A } { \cos A - \sin A } \quad A \neq \frac { ( 2 n + 1 ) \pi } { 4 } \quad n \in \mathbb { Z }$$ (b) Hence solve, for \(0 \leqslant \theta < 2 \pi\) $$\sec 2 \theta + \tan 2 \theta = \frac { 1 } { 2 }$$ Give your answers to 3 decimal places.

10. The amount of an antibiotic in the bloodstream, from a given dose, is modelled by the formula $$x = D \mathrm { e } ^ { - 0.2 t }$$ where \(x\) is the amount of the antibiotic in the bloodstream in milligrams, \(D\) is the dose given in milligrams and \(t\) is the time in hours after the antibiotic has been given. A first dose of 15 mg of the antibiotic is given.

1. Use the model to find the amount of the antibiotic in the bloodstream 4 hours after the dose is given. Give your answer in mg to 3 decimal places. A second dose of 15 mg is given 5 hours after the first dose has been given. Using the same model for the second dose,
2. show that the total amount of the antibiotic in the bloodstream 2 hours after the second dose is given is 13.754 mg to 3 decimal places. No more doses of the antibiotic are given. At time \(T\) hours after the second dose is given, the total amount of the antibiotic in the bloodstream is 7.5 mg .
3. Show that \(T = a \ln \left( b + \frac { b } { \mathrm { e } } \right)\), where \(a\) and \(b\) are integers to be determined.